Journal of Hydraulic Engineering

The development of tidal flow modeling must be seen within the context of its applications (especially to morphological and water quality problems), and then to its integration with short-period wave modeling, cohesive and noncohesive sediment process modeling, and a wide range of biochemical and ecological process modeling activities. Moreover, these models are themselves dependent on extensive measuring programs carried out both in the field and in the laboratory. The application of tidal models to a number of very large projects has justified the correspondingly rapid development of both integrated modeling capabilities and field measuring capabilities, together with a continuing investment in laboratory studies. Alongside this, the ongoing large-scale dissemination of fourth-generation modeling tools has provided a further impetus to enhance the reliability, flexibility, and availability of generic tidal modeling systems. The range of tidal modeling is introduced here within this context and in relation to the working environments within which tidal models are now most commonly applied.

In this paper we present a brief overview, with a comprehensive set of references to recently published papers for detail, of a range of techniques used in tidal hydrodynamic modeling and recent progress in the field. Although the paper aims to introduce the topic to someone new to the area, the main emphasis is on new developments in near coastal (shallow sea) tidal modeling, and consequently we concentrate on three-dimensional modeling, which has progressed significantly in recent years. The paper gives a brief account of methods used to discretize the hydrodynamic equations in the horizontal, namely finite difference, finite element and boundary-fitted coordinates, and through time. The representation of tidal current structure in a three-dimensional model using a spectral method in the vertical is also described. In this method the subgrid scale vertical diffusion of momentum, which has a major influence on tidal current profiles, is parameterized using a flow-dependent eddy viscosity. Recent work on the chaotic behavior of particles released into two-dimensional tidal flow models and the use of adjoint models is also briefly considered.

The second part of this review deals with three-dimensional tidal models for homogeneous and stratified sea regions in which the vertical eddy viscosity and diffusivity are computed from turbulence energy submodels. The formulation of a range of turbulence energy models is presented, and the solution of the three-dimensional equations using finite difference methods in the vertical on a number of transformed grid (log, log-linear) is described. The application of turbulence energy models is illustrated using results from a three-dimensional model of tides in the Irish Sea. A comparison of computed tidal current profiles determined using a two-equation turbulence closure model with those derived using a simple flow-dependent eddy viscosity model, and with observations, did not reveal any significant differences between the models. Recent work on the effect of stable stratification on turbulence energy and tidal currents is also considered, as this is the more critical test of the accuracy of turbulence energy models. Processes influencing the internal tide generation in regions of steep topography are also examined using the three-dimensional turbulence energy model with a prognostic density field. The role of the nonlinear terms in the generation of short internal waves at the top of the shelf break, and intense surface and bed mixing in this region is examined together with the effects of an upwelling and downwelling favorable wind.

The paper describes an accurate numerical scheme for simulating two-dimensional solute transport in coastal and estuarine waters. The operator-splitting algorithm and a conventional discretization (i.e., no splitting) algorithm have been used to solve the solute transport (or advective-diffusion) equation. To avoid the occurrence of numerical oscillations, the ULTIMATE algorithm and the third-order accurate QUICKEST scheme have been used to represent the advective terms in the equation. A boundary-fitted curvilinear coordinate system has also been deployed to obtain a more accurate solution of the tidal currents. The model has been verified for a series of numerical tests, and has been applied to predict the water quality constituent and sediment transport fluxes in the Humber Estuary, U.K.

The field of storm surge modeling has developed and matured considerably over the past 30 years. Several operational surge models have been implemented in Europe over the past decade. These operational advances have been accompanied by research into new areas, chiefly coupled surge-wave models and data assimilation, both of which are covered in this review. Surge-wave models attempt to provide more realistic model physics for the crucial area of air-sea interaction. Assimilation techniques use available data to generate improved numerical solutions, despite model inadequacies. The current activity in surge modeling and related areas is highlighted by the large proportion of recent papers cited in this work. Despite this activity, there is a general recognition that inadequate meteorological inputs remain the weak link in surge modeling. In addition, the advances in midlatitude modeling are not mirrored in the tropics. This is largely due to difficulties in predicting the paths and properties of tropical storms, but it also serves to emphasize the importance that has been attached to storm surge modeling in Europe.

A review is given of mathematical and computer modeling of many of the physical processes that effect the dilution of tracers released in tidal flows: tidal trapping, differences across the flow in tidal speeds, density currents, flow separation. The modeling approaches given most emphasis are tidal trapping, shear dispersion, chaotic stirring, and particle tracking. The central theme of this review is that during the first few tidal cycles, stretching, twisting, and splitting of the tracer patch gives rapidly increasing rates of dilution. There is a water depth (about 30 m) or estuary width (about 200 m) at which the tidal contribution to the eventual dilution rate is largest (about 3 km² d⁻¹ or 35 m² s⁻¹). Careful selection of the position, timing, and release rate of a discharge can allow a given amount of effluent to be diluted much more rapidly and to have a much reduced environmental impact.

Analytical theory is used to describe the tidal constituents generated in suspended sediment concentrations of clay, silt, and sand eroded by M₂ and S₂ tidal currents. The associated model involves only the fall velocity W_s, the vertical dispersion coefficient E (assumed constant), and water depth D. For W_s> 10 E/D particles remain close to the bed, whereas for W_s< 0.1 E/D particles are well mixed vertically. For W_s>E/D, deposition occurs by advective settling at a rate approximating (1/2)W_sC_z=0 where C_z=0 is the concentration at the bed. Conversely for E>>DW_s settlement is more dependent on the dispersion coefficient E. A deposition rate −α$\overline{C}$($\overline{C}$ depth-averaged concentration) is derived by assuming an exponential decay rate e^−αt, with typical half-lives of 1 min to 2 h for sand and from 1 to 100 d for clay. In shelf seas, the suspended sediment concentration is likely to be determined by the silt-clay fraction with enhanced mean and MSf constituents compared to the associated diurnal and semidiurnal constituents. Opposition of the M₄ and MS₄ constituents at neap tide may significantly reduce the anticipated quarter-diurnal component of sediment concentrations.

The relationship between profile shape and wave forcing at shores composed of soft mud is examined. At a given still water level, the modality of change in profile shape under wave action depends on the fluidization potential of mud, which in turn is contingent on the wave characteristics and the dissipative properties of mud. An important determinant of profile shape in areas where mud can be fluidized is the wave attenuation coefficient. Comparatively large values of this coefficient tend to correlate with erosional, concave-upward profiles, and low values with accretionary, convex-upward shapes. Assuming that the rate of energy dissipation per unit area of nonbreaking, normally incident waves is constant in the nearshore shallow zone, an equation to simulate soft-mud shore profiles has been developed. This equation is shown to be applicable to profile data obtained from a variety of mud shore environments. However, it is noted that realistic simulations of the accretionary and erosional characteristics of mud profiles under changing hydrodynamic conditions will require considerations of cross-shore as well as alongshore sediment sources and sinks.

Residual baroclinic current systems are of fundamental importance in estimating long-term mass transport. In most estuaries, vertical and longitudinal density variations govern the flow patterns and transport processes. In the case of a strait-basin system, especially one that has a large rounded basin greater than the Rossby deformation radius, the lateral density variations should be included as well. This means that the earth's rotation and the stratification of the water become predominant factors. In the present study, residual current data observed in Osaka Bay are examined and are compared to the results of numerical simulation of the three-dimensional (3D) baroclinic flow. Tideand density-induced residual circulations and a tidal front were confirmed and their generating mechanisms were clarified. Furthermore, 3D Lagrangian trajectory computation demonstrated that the vertical estuarine circulation causes an anticyclonic circulation appearing in the head of a rounded bay, which is generated by the horizontal divergence of the upper layer water under the effects of the earth's rotation. It is a peculiarity when compared to many other estuarine and coastal waters dominated by estuarine circulation.